Question: For any T ≥ 0, denote by τT (ω) the first passage time of the MBM to the origin after time T; that is, τT (ω) = inf{s ≥ T | w(s, ω) = 0}. Show that the stochastic equation dx = 3x1/3 dt + 3x2/3 dw, with the initial condition x(0) = 0, has infinitely (unaccountably) many solutions of the form
This example is due to Itô and Watanabe
Next, we consider a system of Itô equations of the form
where wj (t) are independent MBMs and x = (x1, x2, . . ., xn). If the coefficients satisfy a uniform Lipschitz condition, the proofs of the existence and uniqueness theorem and of the localization principle are generalized in a straightforward manner to include the case of systems of the form (4.10).